Examples 7.1.9(a):

Show that the constant function f(x) = c is Riemann integrable on any interval [a, b] and find the value of the integral.
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We have to compute the upper and lower sums for an arbitrary partition, then find the appropriate inf and sup to compute the lower and upper integrals. If they agree, we are done and the common value is the answer. So, here we go:

Take an arbitrary partition P = {x₀, x₁, ..., x_n}. The lower sum of f(x) = c is:

L(f, P) = c (x₁ - x₀) + c (x₂ - x₁) + ... + c (x_n - x_n-1)
= c (x_n - x₀) = c (b - a)

because the inf over any interval (as well as the sup) is always c, and the above sum is telescoping.

Similarly, we have that

U(f, P) = c (b - a)

Hence, the upper and lower sums are independent of the particular partition. Therefore f is integrable and

I^*(f) = I_*(f) = c (b - a)

In particular,

f(x) dx = c (b - a)

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007